MEN OF MATHEMATICS with respect to a? is equal to the function itself - that is, & is the only function which is equal to its derivative.* The concept of transcendence* is extremely simple, also extremely important. Any root of an algebraic equation whose coefficients are rational integers (0, ~1» i2, . - -) is called an algebraic number. Thus V — 1, 2-TS are algebraic numbers, because they are roots of the respective algebraic equations #3 j^ i — o, 5Qx — 139 = 0, in which the coefficients (1,1 for the first; 50, - 139 for the second) are rational integers. A 'number' which is not algebraic is called transcendental. Other- wise expressed, a transcendental number is one which satisfies no algebraic equation with rational integer coefficients. Xow, given any 'number' constructed according to some definite law, it is a meaningful question to ask whether it is algebraic or transcendental. Consider, for example, the follow- ing simply defined number, _i +JL J.1 -i- * -L- 1 4- 10 102 ' 10* ' 10-* ' 10120 "" * * in which the exponents 2, 6, 24, 120, ... are the successive 'factorials', namely 2 = 1x2, 6 = 1x2x3, 24 =1x2x3 X 4,120 = 1x2x3x4x5,..., and the indicated series continues 6to infinity* according to the same law as that for the terms given. The next term is —- ; the sum of the first three terms is •! + -01 -f -000001, or -110001, and it can be proved that the series does actually define some definite number which is less than -12. Is this number a root of any algebraic equation with rational integer coefficients? The answer is no, although to prove this without having been shown how to go about it is a severe test of high mathematical ability. On the other hand, the number defined by the infinite series --h-4- —+ — 4- 10* 103 1011 1014 * Strictly, ae*t where a does not depend upon #, is the most general, but the 'multiplicative constant* a is trivial here. 510